What does this allow the attacker to do? Steal data? The post isnt very clear.
It's stochastic; the attacker randomly gets data from whatever happens to be using the XMM/YMM/ZMM registers at the time. So if the attacker could eavesdrop in the background constantly, they might eventually see a password. Or they might be able to trigger some system code that processes your password, then eavesdrop for the next few milliseconds.
The attacker needs to run code on your machine. Unclear if running code in a web browser is sufficient or not. It requires an unusual sequence of machine instructions, which isn't necessarily possible in JS/WASM, but 'sounds' says they did it: https://news.ycombinator.com/item?id=36849767